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Bipolar diffusion charging of aerosol particles—I: experimental results within the diameter range 4–30 nm
J. Aerosol Scl., Vol, 14. No. 5. p p 671 ~o77. 1983. Printed in Great Britain.
t
0021-8502/83 $ 3 0 0 + 0 . 0 0 1983 Pergamon Press Ltd
BIPOLAR D I F F U S I O N C H A R G I N G OF AEROSOL PARTICLES--I: EXPERIMENTAL RESULTS WITHIN THE DIAMETER R A N G E 4-30 nm* A. HussIN, H. G. SCHEIBEL,K. H. BECKER and J. PORSTEND()RFERf Isotopenlaboratorium der Universit~it G6ttingen, Burckhardtweg 2, D-3400 G6ttingen, F.R.G. (Received 24 March 1983) Abstract--Experimental results obtained on the fraction of charged particles of a monodisperse aerosol with a diameter of less than 10 nm can help to decide which of the charging theories most correctly describes the charging process in the whole particle size range. In this study the experimental results on bipolar diffusion charging of aerosol particles with a diameter between 4 and 30 nm are presented. A difference between the fraction of positively and negatively charged aerosol particles in equilibrium was measured, using a bipolar charger with equal negative and positive ion concentrations. The results were in best ag/'eement with the bipolar charging theory of Fuchs, taking into account the different values of the physical parameters for negative and positive ions.
INTRODUCTION The diffusion charging of aerosol particles by unipolar and bipolar ions has been extensively studied on a theoretical basis. The various theories proposed describe with quite good agreement the charging process in particles with diameters exceeding 30 nm. However, from these theories significantly different probabilities in charging of particles less than 10 nm are predicted. In Fig. 1 the calculated probabilities of charging of Fuchs (1963), Gentry and Brock (1967) and Keefe et al. (1959) (Boltzmann charge distribution) are compared with the measured values of Nolan and Kennan (1949), Pollak and Metnieks (1962), Liu and Pui
~N
~
~ o o [3 O
10-1
" ! ,10-2
~'uch$
O
O •
Nolan ane Kennon lg4.! Pollak and ~ t n l e k $
D
LIU ond PU~ 1974 =
1962 Bol'~zmonn
&
,10-3
10 100 particle diameter d [nm]
1000
Fig. 1. Charging probabilities of particles in bipolar charge equilibrium as a function of the particle diameter.
* Part of this work was presented at the 10th Annual Conference of the Gesellschaft fiir Aerosolforschung, Bologna, September 1982. ÷ Person to whom correspondence should be addressed. 671
672
a
HL'SSIN et al.
(1974) and Kojima (1978). The values measured by Pollak and Metnieks (1962) and Kojima (1978) for particle sizes between 8 and 20 nm suggest that Boltzmann's law does not correctly describe the charge distribution for nanometer particles. Only precisely measured results tor diameters less than 10 nm can provide evidence as to which theory gives an adequate description of the charging process in the whole particle size range. One of the principal difficulties confronting experimental aerosol physicists in the past has been the lack of a suitable means of generating monodisperse aerosols in the diameter range 20 nm with high particle concentrations. Therefore the first important step l\~r this investigation was the construction of an aerosol generator, which produces monodisperse particles in this submicron particle size range (Scheibel and Porstend6rfer, 1983). The newly developed condensation type aerosol generator, together with electrostatic classification (Whitby et al., 1972), generates monodisperse Ag-aerosols in the range of diameters between 2 and 300 nm. The particle concentrations vary from 103-106 particles/cm 3 depending on particle size, and the standard deviation of the number size distribution is about 12 ",, All the particles of the produced aerosol are positively or negatively charged with one elementary charge, and therefore the particle concentration can be measured easily by means of an aerosol filter electrometer (Hewitt, 1957; Knutson and Whitby, 1975). The first experimental results on charging probability in this small particle size range were obtained during a workshop in Vienna (Reischl et al., 1983). However, in these experiments the difference between charging probabilities of negatively and positively charged aerosol particles and the influence of the negative and positive ion concentrations as a charger were not considered. In this study the experimental results of charging probabilities for negatively and positively charged particles in a bipolar atmosphere are reported. Different values of the fraction of positively and negatively charged particles of an aerosol in bipolar charge equilibrium were measured under the experimental condition of equal negative and positive ion concentrations in the bipolar ion source,
EXPERIMENTAL METHOD The experimental system for the determination of the charging probability of aerosols is shown in Fig. 2. The measurements were made in two stages. Firstly the singly charged aerosol of the generator was drawn through a second electrostatic aerosol classifier (the first electrostatic aerosol classifier is part of the aerosol generator) without using the bipolar ion source. By measuring the current produced by the singly charged particles with an aerosol filter electrometer the concentration of the number of aerosol particles was determined. Secondly a bipolar charger (IS) was used in addition to establish an equilibrium bipolar charge distribution of the monodisperse aerosol and the first measurement was repeated twice: one measurement of the negative and one of the positive current, produced by the negatively and positively charged fraction of aerosol particles, respectively. In Fig. 3 an example of the measured current distributions A I z / A V, AI +/A V and AI - / A V is shown. As the maxima of current for the three measurements are located at the same voltage, and as there is no second maximum the particles of the charged fractions (positive and negative) have the same mobility, i.e. they are singly charged. Particles with two charges have to be found at voltages 1/2 of the current maxima, but these concentrations are below the sensitivity of the measurement. From these three measurements it is possible to calculate the fractions of the singly positive (N ~/Z) and negative (N ~-/Z)charged particles in equilibrium (Hussin, 1983)
.N~
ZANy(d) d
Z
ZAI+--(V) )
~ AZ(d) d
~AIz(V) V
(I)
Bipolar diffusion charging of aerosol particles--I .
.
MONODISPERSE AEROSOL GENERATOR . . . . . . . . . . . . . . . . . . . . , AIR
.
.
.
.
.
673
. AIR
,V V
BIPOLAR ION SOURCE
F~M
~r . . . . . . .
O~ rr e
nr
rI li
FM
~
1
F
~
,
V
_
II
m
i--I
I
.
'
ELECTROMETERI
)F
I J
..........................
Fig. 2. Arrangement for measuring the charging probability (V: valve, FM: flowmeter, F: filter, FU: furnace, IS: bipolar ion source, HV: high voltage, PS: power supply, E: equivalent volume).
with the aerosol concentration
z = No + N; + N? = Z A Z (a).
(2)
d
This experimental method based on the measurement o f three current distributions with the same instrumentation and procedure, allow an accurate determination o f the fractions of charged particles of an aerosol.
particle diameter [nm] 1008 127 1/,,5 16 175 19 201 20.8 o AIz/Av • AI-/AV
• AI+/AV
10-13 o. E
10"I~.
'.: \~',/
O 10-~6
so
7b ~0 ~0 ~0 ~0
I~o 19o
classification voltage IV]
Fig. 3. The measured current distributions of the aerosol in bipolar charge equilibrium.
A HI:SSlY et a l
674
BIPOLAR
ION
SOURCE
There are several physical parameters which influence the charging process. These include the mobility and mass of the ions, as well as their mean molecular velocity, diffusion coefficient and mean free path of the ions. Not all of these parameters are independent, however, some being related to each other by fundamental physical laws. A number of experimental studies show that the electrical mobility of the negative ions is greater than that of the positive ions in air (Mohnen, 1977). Therefore the different physical parameters of the negative and positive ions, engaged in the charging process, influence the ion concentration in the ion source and the charging probability of the aerosol. The positive (n +) and negative (n-) ion concentration in a bipolar ion source is described by the following equation dn*+ -q-~,n-n -fi±n ± z - a ~ n ±. dt
(3)
One part of the ions disappears by recombination (a~ n - n +), while the other fraction disappears by attachment on walls (x,n -+) of the ion source and on aerosol particles (#± Zn ± ). The recombination coefficient a, for this is 1.6 x 1 0 - 6 c m 3 sec-t (Bricard, 1965), #± is the total attachment coefficient of ions to aerosols, Z the aerosol concentration, and a~ the attachment rate to the walls of the ion source, q is the production rate of the ions. In this study an ion source was built, which resulted in equal concentrations of positive and negative ions in the charger (n- = n ÷). This can be obtained if the recombination rate (~, n - n +) is much greater than the attachment processes (fi ± n -+Z + a~ n -+). In our experiments we used the x-radiation of Am-241, covered on the walls inside a tube. as bipolar ion source. The radioactivity was about 6 mCi. The measured ion production rate was 3.6 x 101° ions cm -3 sec -t. The ion concentrations in the source of about 1.5 x 108 ions cm- 3 were so high, that the concentrations of the positive and negative ions were equal, resulting in different charging probabilities of aerosols, caused by the different physical parameters of the positive and negative ions.
E X P E R I M E N T A L RESULTS AND C O M P A R I S O N WITH T H E O R I E S The measured values of the fraction of the positively ( N ~ / Z ) a n d negatively (Ni-/Z) charged particles as a function of the particle diameter are shown in Fig. 4. There is a rapid
I-rY
i0_~ ~.)
LLI
~ 10-2 LL 0
/' /so~
r /
zmonn
//Gen,ry
< ta-
/, /
Z F--
10-3
I / ; ,l,
/
o t~gjgtive tEXPERIMENTS g • pos~t,ve - r HEORIES
t0 PARTICLE DIAMETER
100 d. nm
Fig. 4. The measured and theoretical fractions of charged particles of monodisperse aerosols in a bipolar ion atmosphere with ion concentrations n - = n*.
Bipolar diffusioncharging of aerosol particles--I
675
decrease of the number of the charged particles for smaller sizes. Only 5 ~o of the 4.5 nm particles are electrically charged. The measured fraction of the negatively charged particles is about 75 oo greater than the positively charged ones. The charge equilibrium of the aerosol was established by means of an ion atmosphere with n + = n-. Figure 4 includes the theoretical curves of Fuchs (1963), Gentry and Brock (1967) and the Boltzmann charge distribution. The Boltzmann approach to the charging theory, as suggested by some authors, predicts much smaller fractions of charged particles for diameters below 10 nm in contrast to our experimental data. Furthermore the difference between the charging probabilities for positively and negatively charged aerosols cannot be described by the Boltzmann charging formula. Figure 4 also shows that our experimental results are in better agreement with the theory of Fuchs (1963) than with the other proposed theories. Since no distinction between the properties of positive and negative ions is made by Fuchs, we calculated the charging probabilities for negatively and positively charged aerosol particles, taking into account the different values of mass, mobility, diffusion coefficient, mean thermal velocity and mean free path of the ions. The charge acquisition process is formulated as a stochastic birth-death process and can be described by the following set of simultaneous differential equations: dN~ n-+ + + dt -f l ~ P - I N f - I +n-V fll,p+ ~ l N:+ l
-
n+
fl2,pNp ± -+ - n
+ f l l+, , N ~ +
(4)
in which N~ represents the number concentration of positive and negative aerosol particles of equal size, carrying p elementary charges. Then fl-+ 2~, is the attachment coefficient of ions ( + ) on particles having p charges of the same sign, and fl~,, the corresponding coefficient between the same small ions and aerosol particles having p charges of opposite sign. An approximation of the charge distribution under the steady state condition d N ~ / d t = 0 from equation (4) can be obtained by assuming - =
= fiG,
/V,,
(5)
as often reported (Keefe et al., 1959; Fuchs, 1963; etc.). In this case Ni ~ = Ni-', N~ = N; ..... N; = N;. However, the experimental results show that there is a difference between the concentration of positively and negatively charged particles (N~- 4: N~- ). A more general solution of the equation system (4) can be obtained, when the number of particles of one size with higher charges than p = m is neglected (N,~+I = N~,÷2 . . . . = 0). Then a system of 2 m + l equations gives the relation N~ N~_x
= ---=n -+ n+
fiE,-, n+
+
- - ~lap+l
(6) Ny+x/Np
or --
P
N: = H [A? ] No
(7)
j= 1
with
A f = n---~- fll~J + (nZ/n.v) ~2"~J + -- ~ , ~, j + l U ? + l / U ~
(8)
With the total particle concentration of the monodisperse aerosol Z=N o+ ~ (N~+N~-) p=l
(9)
676
A
-~
Hussl~ ct at
Z
1
lOli I
j J/ ,//" oo,
I
I0
i i 100
PARTICLE DIAMETER d.nm
Fig. 5. Comparison of the measured fractions of charged particles of monodisperse aerosols with values calculated from Fuchs's charging theory. The ion concentrations in the bipolar charger are n = n ÷. (Ion parameters: m- = 101ainu, m÷ = 140ainu, D- = 0.036cm~/sec, D + = 0.029cm2/sec. c- = 2.5 x 104era/see, c ÷ = 2.1 x 104cm/sec, ,;.- = 1.7 x 10-6cm, 2 ÷ = 1.4 × | 0 - 6 c m . ) the charge distribution is P
1-I [A?]
N~_ =
j=l p=l
IJ
=1
(10) j=l
t
By means of equation (10), we calculated N ~ / Z taking into account the ion condition in our ion source n - = n +. The attachment coefficients fl,~,p, #~p and #o~ were calculated according to the charging theory o f Fuchs (1963). As shown in Fig. 5, the best agreement with the experimental results was obtained, assuming the diffusion constants D - = 0.036 cm2/sec and D + = 0.029 cmZ/sec, the masses m - = 101 amu and m + = 140 ainu and the derived parameters as mean thermal velocities c - = 2.5 x 104 cm/sec and c ÷ = 2.t × 104 cm/sec and the mean free paths 2 - = 1.7 x 10 -6 cm and 2 ÷ = 1.4 × 10 -6 cm for negative and positive ions, respectively.
SUMMARY In summary this study reports experimental results of the fractions of negatively and positively charged particles o f a monodisperse aerosol in the particle size range below 30 nm, equilibrium being established by means o f a bipolar ion source with equal negative and positive ion concentrations. The a m o u n t of the charged particles ranges from a b o u t 40 5~, for 30 nm particles to about 5 % for the smallest particles (d = 4.5 nm) with a significant difference between the fractions of positively and negatively charged fractions of about 75 30. The experimental results can be described by means of the charging theory of Fuchs, if realistic assumptions for the different ion properties are used and if different numbers of charged particles (N~ :~ Np-) are taken into account in the solution of the charging equations. In order to clarify the influence of different ion concentrations in a charger on the charging probabilities in charge equilibrium, further experimental studies are required. Experimental results in the larger aerosol diameter range between 0.1 and 2tLm (Porstend6rfer et al.) agree well with the theory of Fuchs.
REFERENCES Bricard, J. (1965) Problems of Atmospheric and Space Electricity (Edited by Coroniti. S. C.), p. 82. Elsevier Publishing
Company, Amsterdam.
Bipolar diffusion charging of aerosol particles--I
677
Fuchs, N. A. t1963) Geo[is. Pura Appl. 56, 185. Gentry, J. and Brock, J. R. (1967) d. Chem. Phys. 47(1), 64. Hussin, A. (1983) Ph.D. Thesis, Faculty of Physics, University of Giessen. F.R.G Hewitt, G. W. {1957) Am. J. Electr. Engng. 76, 300. Keefe, D., Nolan, P. J. and Rich, T. A. (1959) Proc. R.I.A. 60A, 27. Knutson, E O. and Whitby, K. T. (1975) J. Aerosol Sci. 6, 443. Kojima, H. (1978) Atmospheric Environment 12, 2363. Liu, B. Y. H. and Pui, D. Y. H. (1974) J. Colloid Interface Sci. 49, 305. Mohnen, V. A. I 1977) Electrical Processes in Atmospheres (Edited by Dolezalek, H. and Reiter R.), p. 1. D. Steinkopff, Darmstadt. Nolan, P. J. and Kennan, E. L. (1949) Proc. R. Jr. Acad. 52A, 171. Pollak, L. W. and Metnieks, A. L. (1962) Geofis. Pura Appl. 53, 111. Porstend6rfer, J., Hussin, A., Scheibel, H. G. and Becket, K. H. (in preparation). Porstend6rfer, J. and Mercer, T. T. (1977) Health Physics 37, 191. Reischl, G. P., Scheibel, H. G. and Porstend6rfer, J. (1983) J. Colloid InterJace Sci. 91,272. Scheibel, H. G. and Porstend6rfer, J. (1983) J. Aerosol Sci. 14. Whitby, K. T., Liu, B. Y. H., Husar, R. B. and Barsic, N. J. (1972) J. Colloid Interface Sci. 39, 136.
t
0021-8502/83 $ 3 0 0 + 0 . 0 0 1983 Pergamon Press Ltd
BIPOLAR D I F F U S I O N C H A R G I N G OF AEROSOL PARTICLES--I: EXPERIMENTAL RESULTS WITHIN THE DIAMETER R A N G E 4-30 nm* A. HussIN, H. G. SCHEIBEL,K. H. BECKER and J. PORSTEND()RFERf Isotopenlaboratorium der Universit~it G6ttingen, Burckhardtweg 2, D-3400 G6ttingen, F.R.G. (Received 24 March 1983) Abstract--Experimental results obtained on the fraction of charged particles of a monodisperse aerosol with a diameter of less than 10 nm can help to decide which of the charging theories most correctly describes the charging process in the whole particle size range. In this study the experimental results on bipolar diffusion charging of aerosol particles with a diameter between 4 and 30 nm are presented. A difference between the fraction of positively and negatively charged aerosol particles in equilibrium was measured, using a bipolar charger with equal negative and positive ion concentrations. The results were in best ag/'eement with the bipolar charging theory of Fuchs, taking into account the different values of the physical parameters for negative and positive ions.
INTRODUCTION The diffusion charging of aerosol particles by unipolar and bipolar ions has been extensively studied on a theoretical basis. The various theories proposed describe with quite good agreement the charging process in particles with diameters exceeding 30 nm. However, from these theories significantly different probabilities in charging of particles less than 10 nm are predicted. In Fig. 1 the calculated probabilities of charging of Fuchs (1963), Gentry and Brock (1967) and Keefe et al. (1959) (Boltzmann charge distribution) are compared with the measured values of Nolan and Kennan (1949), Pollak and Metnieks (1962), Liu and Pui
~N
~
~ o o [3 O
10-1
" ! ,10-2
~'uch$
O
O •
Nolan ane Kennon lg4.! Pollak and ~ t n l e k $
D
LIU ond PU~ 1974 =
1962 Bol'~zmonn
&
,10-3
10 100 particle diameter d [nm]
1000
Fig. 1. Charging probabilities of particles in bipolar charge equilibrium as a function of the particle diameter.
* Part of this work was presented at the 10th Annual Conference of the Gesellschaft fiir Aerosolforschung, Bologna, September 1982. ÷ Person to whom correspondence should be addressed. 671
672
a
HL'SSIN et al.
(1974) and Kojima (1978). The values measured by Pollak and Metnieks (1962) and Kojima (1978) for particle sizes between 8 and 20 nm suggest that Boltzmann's law does not correctly describe the charge distribution for nanometer particles. Only precisely measured results tor diameters less than 10 nm can provide evidence as to which theory gives an adequate description of the charging process in the whole particle size range. One of the principal difficulties confronting experimental aerosol physicists in the past has been the lack of a suitable means of generating monodisperse aerosols in the diameter range 20 nm with high particle concentrations. Therefore the first important step l\~r this investigation was the construction of an aerosol generator, which produces monodisperse particles in this submicron particle size range (Scheibel and Porstend6rfer, 1983). The newly developed condensation type aerosol generator, together with electrostatic classification (Whitby et al., 1972), generates monodisperse Ag-aerosols in the range of diameters between 2 and 300 nm. The particle concentrations vary from 103-106 particles/cm 3 depending on particle size, and the standard deviation of the number size distribution is about 12 ",, All the particles of the produced aerosol are positively or negatively charged with one elementary charge, and therefore the particle concentration can be measured easily by means of an aerosol filter electrometer (Hewitt, 1957; Knutson and Whitby, 1975). The first experimental results on charging probability in this small particle size range were obtained during a workshop in Vienna (Reischl et al., 1983). However, in these experiments the difference between charging probabilities of negatively and positively charged aerosol particles and the influence of the negative and positive ion concentrations as a charger were not considered. In this study the experimental results of charging probabilities for negatively and positively charged particles in a bipolar atmosphere are reported. Different values of the fraction of positively and negatively charged particles of an aerosol in bipolar charge equilibrium were measured under the experimental condition of equal negative and positive ion concentrations in the bipolar ion source,
EXPERIMENTAL METHOD The experimental system for the determination of the charging probability of aerosols is shown in Fig. 2. The measurements were made in two stages. Firstly the singly charged aerosol of the generator was drawn through a second electrostatic aerosol classifier (the first electrostatic aerosol classifier is part of the aerosol generator) without using the bipolar ion source. By measuring the current produced by the singly charged particles with an aerosol filter electrometer the concentration of the number of aerosol particles was determined. Secondly a bipolar charger (IS) was used in addition to establish an equilibrium bipolar charge distribution of the monodisperse aerosol and the first measurement was repeated twice: one measurement of the negative and one of the positive current, produced by the negatively and positively charged fraction of aerosol particles, respectively. In Fig. 3 an example of the measured current distributions A I z / A V, AI +/A V and AI - / A V is shown. As the maxima of current for the three measurements are located at the same voltage, and as there is no second maximum the particles of the charged fractions (positive and negative) have the same mobility, i.e. they are singly charged. Particles with two charges have to be found at voltages 1/2 of the current maxima, but these concentrations are below the sensitivity of the measurement. From these three measurements it is possible to calculate the fractions of the singly positive (N ~/Z) and negative (N ~-/Z)charged particles in equilibrium (Hussin, 1983)
.N~
ZANy(d) d
Z
ZAI+--(V) )
~ AZ(d) d
~AIz(V) V
(I)
Bipolar diffusion charging of aerosol particles--I .
.
MONODISPERSE AEROSOL GENERATOR . . . . . . . . . . . . . . . . . . . . , AIR
.
.
.
.
.
673
. AIR
,V V
BIPOLAR ION SOURCE
F~M
~r . . . . . . .
O~ rr e
nr
rI li
FM
~
1
F
~
,
V
_
II
m
i--I
I
.
'
ELECTROMETERI
)F
I J
..........................
Fig. 2. Arrangement for measuring the charging probability (V: valve, FM: flowmeter, F: filter, FU: furnace, IS: bipolar ion source, HV: high voltage, PS: power supply, E: equivalent volume).
with the aerosol concentration
z = No + N; + N? = Z A Z (a).
(2)
d
This experimental method based on the measurement o f three current distributions with the same instrumentation and procedure, allow an accurate determination o f the fractions of charged particles of an aerosol.
particle diameter [nm] 1008 127 1/,,5 16 175 19 201 20.8 o AIz/Av • AI-/AV
• AI+/AV
10-13 o. E
10"I~.
'.: \~',/
O 10-~6
so
7b ~0 ~0 ~0 ~0
I~o 19o
classification voltage IV]
Fig. 3. The measured current distributions of the aerosol in bipolar charge equilibrium.
A HI:SSlY et a l
674
BIPOLAR
ION
SOURCE
There are several physical parameters which influence the charging process. These include the mobility and mass of the ions, as well as their mean molecular velocity, diffusion coefficient and mean free path of the ions. Not all of these parameters are independent, however, some being related to each other by fundamental physical laws. A number of experimental studies show that the electrical mobility of the negative ions is greater than that of the positive ions in air (Mohnen, 1977). Therefore the different physical parameters of the negative and positive ions, engaged in the charging process, influence the ion concentration in the ion source and the charging probability of the aerosol. The positive (n +) and negative (n-) ion concentration in a bipolar ion source is described by the following equation dn*+ -q-~,n-n -fi±n ± z - a ~ n ±. dt
(3)
One part of the ions disappears by recombination (a~ n - n +), while the other fraction disappears by attachment on walls (x,n -+) of the ion source and on aerosol particles (#± Zn ± ). The recombination coefficient a, for this is 1.6 x 1 0 - 6 c m 3 sec-t (Bricard, 1965), #± is the total attachment coefficient of ions to aerosols, Z the aerosol concentration, and a~ the attachment rate to the walls of the ion source, q is the production rate of the ions. In this study an ion source was built, which resulted in equal concentrations of positive and negative ions in the charger (n- = n ÷). This can be obtained if the recombination rate (~, n - n +) is much greater than the attachment processes (fi ± n -+Z + a~ n -+). In our experiments we used the x-radiation of Am-241, covered on the walls inside a tube. as bipolar ion source. The radioactivity was about 6 mCi. The measured ion production rate was 3.6 x 101° ions cm -3 sec -t. The ion concentrations in the source of about 1.5 x 108 ions cm- 3 were so high, that the concentrations of the positive and negative ions were equal, resulting in different charging probabilities of aerosols, caused by the different physical parameters of the positive and negative ions.
E X P E R I M E N T A L RESULTS AND C O M P A R I S O N WITH T H E O R I E S The measured values of the fraction of the positively ( N ~ / Z ) a n d negatively (Ni-/Z) charged particles as a function of the particle diameter are shown in Fig. 4. There is a rapid
I-rY
i0_~ ~.)
LLI
~ 10-2 LL 0
/' /so~
r /
zmonn
//Gen,ry
< ta-
/, /
Z F--
10-3
I / ; ,l,
/
o t~gjgtive tEXPERIMENTS g • pos~t,ve - r HEORIES
t0 PARTICLE DIAMETER
100 d. nm
Fig. 4. The measured and theoretical fractions of charged particles of monodisperse aerosols in a bipolar ion atmosphere with ion concentrations n - = n*.
Bipolar diffusioncharging of aerosol particles--I
675
decrease of the number of the charged particles for smaller sizes. Only 5 ~o of the 4.5 nm particles are electrically charged. The measured fraction of the negatively charged particles is about 75 oo greater than the positively charged ones. The charge equilibrium of the aerosol was established by means of an ion atmosphere with n + = n-. Figure 4 includes the theoretical curves of Fuchs (1963), Gentry and Brock (1967) and the Boltzmann charge distribution. The Boltzmann approach to the charging theory, as suggested by some authors, predicts much smaller fractions of charged particles for diameters below 10 nm in contrast to our experimental data. Furthermore the difference between the charging probabilities for positively and negatively charged aerosols cannot be described by the Boltzmann charging formula. Figure 4 also shows that our experimental results are in better agreement with the theory of Fuchs (1963) than with the other proposed theories. Since no distinction between the properties of positive and negative ions is made by Fuchs, we calculated the charging probabilities for negatively and positively charged aerosol particles, taking into account the different values of mass, mobility, diffusion coefficient, mean thermal velocity and mean free path of the ions. The charge acquisition process is formulated as a stochastic birth-death process and can be described by the following set of simultaneous differential equations: dN~ n-+ + + dt -f l ~ P - I N f - I +n-V fll,p+ ~ l N:+ l
-
n+
fl2,pNp ± -+ - n
+ f l l+, , N ~ +
(4)
in which N~ represents the number concentration of positive and negative aerosol particles of equal size, carrying p elementary charges. Then fl-+ 2~, is the attachment coefficient of ions ( + ) on particles having p charges of the same sign, and fl~,, the corresponding coefficient between the same small ions and aerosol particles having p charges of opposite sign. An approximation of the charge distribution under the steady state condition d N ~ / d t = 0 from equation (4) can be obtained by assuming - =
= fiG,
/V,,
(5)
as often reported (Keefe et al., 1959; Fuchs, 1963; etc.). In this case Ni ~ = Ni-', N~ = N; ..... N; = N;. However, the experimental results show that there is a difference between the concentration of positively and negatively charged particles (N~- 4: N~- ). A more general solution of the equation system (4) can be obtained, when the number of particles of one size with higher charges than p = m is neglected (N,~+I = N~,÷2 . . . . = 0). Then a system of 2 m + l equations gives the relation N~ N~_x
= ---=n -+ n+
fiE,-, n+
+
- - ~lap+l
(6) Ny+x/Np
or --
P
N: = H [A? ] No
(7)
j= 1
with
A f = n---~- fll~J + (nZ/n.v) ~2"~J + -- ~ , ~, j + l U ? + l / U ~
(8)
With the total particle concentration of the monodisperse aerosol Z=N o+ ~ (N~+N~-) p=l
(9)
676
A
-~
Hussl~ ct at
Z
1
lOli I
j J/ ,//" oo,
I
I0
i i 100
PARTICLE DIAMETER d.nm
Fig. 5. Comparison of the measured fractions of charged particles of monodisperse aerosols with values calculated from Fuchs's charging theory. The ion concentrations in the bipolar charger are n = n ÷. (Ion parameters: m- = 101ainu, m÷ = 140ainu, D- = 0.036cm~/sec, D + = 0.029cm2/sec. c- = 2.5 x 104era/see, c ÷ = 2.1 x 104cm/sec, ,;.- = 1.7 x 10-6cm, 2 ÷ = 1.4 × | 0 - 6 c m . ) the charge distribution is P
1-I [A?]
N~_ =
j=l p=l
IJ
=1
(10) j=l
t
By means of equation (10), we calculated N ~ / Z taking into account the ion condition in our ion source n - = n +. The attachment coefficients fl,~,p, #~p and #o~ were calculated according to the charging theory o f Fuchs (1963). As shown in Fig. 5, the best agreement with the experimental results was obtained, assuming the diffusion constants D - = 0.036 cm2/sec and D + = 0.029 cmZ/sec, the masses m - = 101 amu and m + = 140 ainu and the derived parameters as mean thermal velocities c - = 2.5 x 104 cm/sec and c ÷ = 2.t × 104 cm/sec and the mean free paths 2 - = 1.7 x 10 -6 cm and 2 ÷ = 1.4 × 10 -6 cm for negative and positive ions, respectively.
SUMMARY In summary this study reports experimental results of the fractions of negatively and positively charged particles o f a monodisperse aerosol in the particle size range below 30 nm, equilibrium being established by means o f a bipolar ion source with equal negative and positive ion concentrations. The a m o u n t of the charged particles ranges from a b o u t 40 5~, for 30 nm particles to about 5 % for the smallest particles (d = 4.5 nm) with a significant difference between the fractions of positively and negatively charged fractions of about 75 30. The experimental results can be described by means of the charging theory of Fuchs, if realistic assumptions for the different ion properties are used and if different numbers of charged particles (N~ :~ Np-) are taken into account in the solution of the charging equations. In order to clarify the influence of different ion concentrations in a charger on the charging probabilities in charge equilibrium, further experimental studies are required. Experimental results in the larger aerosol diameter range between 0.1 and 2tLm (Porstend6rfer et al.) agree well with the theory of Fuchs.
REFERENCES Bricard, J. (1965) Problems of Atmospheric and Space Electricity (Edited by Coroniti. S. C.), p. 82. Elsevier Publishing
Company, Amsterdam.
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